We are going to use the

Geometer's Sketchpadto graph functions in a way that allows us to change parameters. Aparametermeans a value that may have different values, but doesn't vary once it is set. For example, in the linear function,y= mx+ bmandbareparameters, whileandxareyvariables.Each of the sections below will help you develop your ability to use the Sketchpad to graph functions with dynamic capabilities.

- Open a new sketch. Go to Display >> Preferences; set the Autoshow Label ON for Points (a check must show in the square), and set the Distance units to inches. Go to Graph >> Create Axes. Notice that point A appears at the origin and point B appears at the unit value (1) on the x-axis.
Using the Select tool, explore what happens when you click-&-drag point A? point B?

- Use the Point tool to create a new point, C, NOT on either of the axes. With only point C selected, use Measure >> Coordinates.
What happens as you slide C around?

- Select the coordinates (NOT the point itself!), then use Measure >> Calculate >> Values >> Point C >> x >> OK. What shows on the screen as the measurement?
Repeat this to get the y value of point C to show.

What happens to these values as you move point C around?

Select the measure showing the coordinates as an ordered pair: use Display >> Hide.

- Use the Point tool to create a new point, D, ON the x-axis. With only point D selected, use Measure >> Coordinates. Select the coordinates, then use Measure >> Calculate >> Values >> Point D >> x. WAIT!! BEFORE clicking OK, click the Text Format button at the lower left of the calculator.
How does a measurement in text format show up differently on the screen?

- Activate the Text tool and double-click on the text format measurement. Notice that you can type in anything you'd like, such as "The x value of point D is (leave a space)", or simply "x = ", and click OK. The screen will read, "x = (value)", and the value should change as you move the point around; test that it does.

- You are now ready to make a parameter "slider". Open a new sketch.
Set the Display >> Preferences so that it does NOT Autoshow labels for points. Graph >> Create axes.

- Place a point on the x-axis, away from the unit point. Use the Text tool's black hand to show the label of this point. Double-click on the label and re-name it "X". Measure its coordinates and select them. Use Measure >> Calculate >> Values >> Point X >> x, and click Text format. Use the Text tool to re-label the measurement to read "x = ". Check that the value changes as you slide the point along the x-axis.

- Place a point below the x-axis, under the value of about (—2). Shift+select the point and the x-axis, and Construct >> Parallel line.
Place another point on this new line under the value of about (+2). With only the line selected, use Display >> Hide the line; join the two points with a segment.

- Place an additional point on the segment. Use the Text tool to rename the point as "M". Measure its coordinates; select the coordinates and Calculate >> x in text format. Use the text tool to have the measurement read as "m = ". Check that the measure changes as you slide the point along the segment.
Select and hide (NOT Delete) just the ordered pairs of coordinates.

- Now lets create a function, like
. To do this, open the calculator; select the measurement "m = (value)". The calculator window shows "m". Enter " * " (for multiply), then select the measurement "x = (value)". The calculator window shows "m*x". When you click OK, "m•x = (value)" appears on the sketch.y= mx

- IN ORDER, shift+select "x = (value)" then "m•x = (value)". Use Graph >> Plot as (x, y). A highlighted point should appear in the sketch. If the new point is not immediately obvious, try moving the X point on the x-axis or the M point on the slider to bring it into view.

- IN ORDER, shift+select this new point and the point X on the x-axis. Construct >> Locus.

- What appears should look something like the picture below. That is, there's a line passing through the origin, although your line could have a different slope.

- What happens if you grab-&-drag X along the x-axis? … M along the segment? … one of the endpoints of the slider segment? … the origin point? … the unit point on the axis?

Additional Questions and Extensions

- How would you add the parameter
bto make this a model for studying a general linear equation of the form?y= mx+ b- The calculator has a number of built-in functions, available under the "Functions" button, including the three basic trig functions. How could you illustrate the effects of the parameters a, b and c in the general trig function
y= a•sin(bx) + c ? If you do create this sketch, you might want to go into Display >> Preferences and change the Angle unit to radians. Why?